18.099b Problem Set 4a Due: Thursday, March 11th (in class or before).

Let E ⊂ ℝ be a subset of the real numbers. A point x ∈ E is isolated if there is an open interval containing x but containing no other point of E. That is, there is δ > 0 such that (x − δ, x + δ) ∩ E = {x}. The set E is called discrete if every point in E is isolated.

Write a short exposé that:

• defines discrete sets,

• gives an example (with proof) of a set that is discrete and a set that is not discrete, and

• describes (with proof) which sequences of points in a discrete set are conver-

gent. You are free to organize the material in whatever way you think best. Think of it as a tiny mathematical essay; it should be self-contained and flow smoothly. The exposé must be typeset in Tex.